Work in hyperbolic geometry. Given a triangle , recall that its Saccheri quadrilateral based at is defined as follows: Let be the midpoint of and be the midpoint of . Let be the feet of the perpendiculars from and to respectively.

Continuing with the same notation, suppose now that is an arbitrary point on , and let be a point on the ray with . Show that is also the Saccheri quadrilateral of based at .

HW 6 is due Thursday, April 16 and HW 7 is due Tuesday, April 21, both at the beginning of lecture.

HW6

Work in hyperbolic geometry.

1. Let and be two parallel lines admitting a common perpendicular: There are points and with perpendicular to both and . Suppose that are other points in with , that is, is between and . Let be the foot of the perpendicular from to , and let be the foot of the perpendicular from to .

Show that . That is, and drift apart away from their common perpendicular.

(Note that and are Lambert quadrilaterals, and therefore and . The problem is to show that .)

As an extra credit problem, show that for any number we can find (on either side of ) such that , that is, and not just drift apart but they do so unboundedly.

2. Now suppose instead that and are critical (or limiting) parallel lines, that is, they are parallel, and if and is the foot of the perpendicular from to , then on one of the two sides determined by the line , any line through that forms with a smaller angle than does, cuts at some point.

On the same side as just described, suppose that are points on with , that is, is between and . Let be the foot of the perpendicular from to , and let be the foot of the perpendicular from to .

Show that . That is, and approach each other in the appropriate direction.

As an extra credit problem, show that for any we can choose so that . That is, and are asymptotically close to one another. Do they drift away unboundedly in the other direction?

HW 7

Show that the critical function is continuous. Recall that measures the critical angle, that is, iff there are critical parallel lines and and a point such that if is the foot of the perpendicular from to , and , then and make an angle of measure in the appropriate direction.

(In lecture we verified that is strictly decreasing. This means that the only possible discontinuities of are jump discontinuities. We also verified that approaches as , and approaches as . It follows that to show that has no jump discontinuities, it suffices to verify that it takes all values between and , that is, one needs to prove that for any there is an such that .)

Write a program that receives as input a real symmetric matrix and some tolerance bound , and performs the basic method to generating (and printing) a sequence of matrices until a stage is reached where the entries below the diagonal of are all in absolute value below . Once this happens, the program returns the diagonal entries of as approximations to the eigenvalues of . (Check on a couple of examples that these are indeed decent approximations, at least for of small size and reasonably small values of .)

Most Computer Algebra Systems already have implemented algorithms to find the decomposition of a matrix. Instead of using these pre-programmed algorithms, write your own.

(Turn in the code, plus the couple of examples. Comment your code, so it can be easily understood what you are doing along the way. I’m reasonably familiar with Maple, Mathlab, Sage, and most flavors of C. If you are going to use a different language, please let me know as soon as you can, to see whether it is something I’ll be able to verify or if a different language will be needed instead. Ideally, the user can choose the dimension of the input matrix.)

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]